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The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

DOI: 10.1155/2011/469806

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Abstract:

We consider the Cauchy-Dirichlet problem in for a class of linear parabolic partial differential equations. We assume that is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains. 1. Introduction In this paper, we study the existence and uniqueness of a classical solution to the Cauchy-Dirichlet problem for a linear parabolic differential equation in a general unbounded domain. Let be the differential operator where , , and . The Cauchy-Dirichlet problem is where is an unbounded, open, connected set with regular boundary. In the case of bounded domains, the Cauchy-Dirichlet problem is well understood (see [1, 2] for a detailed description of this problem). Moreover, when the domain is unbounded and the coefficients are bounded, the existence of a classical solution to (1.2) is well known. For a survey of this theory see [3, 4] where the problem is studied with analytical methods and [5] for a probabilistic approach. In the last years, parabolic equations with unbounded coefficients in unbounded domains have been studied in great detail. For the particular case when , there exist many papers in which the existence, uniqueness, and regularity of the solution is studied under different hypotheses on the coefficients; see for example, [6–17]. In the case of general unbounded domains, Fornaro et al. in [18] studied the homogeneous, autonomous Cauchy-Dirichlet problem. They proved, using analytical methods in semigroups, the existence and uniqueness of a solution to the Cauchy-Dirichlet problem when the coefficients are locally , with bounded, and functions with a Lyapunov type growth; that is, there exists a function such that and for some , It is also assumed that has a boundary. Schauder-type estimates were obtained for the gradient of the solution in terms of the data. Bertoldi and Fornaro in [19] obtained analogous results for the Cauchy-Neumann problem for an unbounded convex domain. Later, in [20] Bertoldi et al. generalized the method to nonconvex sets with boundary. They studied the existence, uniqueness, and gradient estimates for the Cauchy-Neumann problem. For a survey of this results, see [21]. Using the theory of semigroups, Da Prato and Lunardi studied, in [22, 23], the realization of the elliptic operator , in the

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